Algebra I : Unit #8 - Our City Statistics: Who We Are and Where We are Going : Lesson #7

Cinderella's Slipper: Scatterplots, Residuals and Goodness of Fit

Objective: SWBAT choose an appropriate function to model a bivariate data set. SWBAT fit a linear function to the data. SWBAT plot, analyze and interpret residuals for a data set.
Standards: HSS-ID.B.6 HSS-ID.B.6a HSS-ID.B.6b HSS-ID.B.6c MP1 MP2 MP3 MP5 MP7
Subject(s): Math
90 minutes
1 Entry Ticket - 15 minutes

At the start of class students will complete Entry Ticket: Cinderella's Slipper where each has to analyze a scatter plot to identify which type of function (linear, exponential or quadratic) would best fit the data. I will also ask students also complete a Turn and Talk to share their ideas and listen to the ideas of a classmate. As they complete the exit ticket I listen for and help students with the use of appropriate statistical vocabulary (MP6: attention to precision). 

Lesson Modification: If computer technology is available, it is possible for students to complete tables and plots for this lesson using technology (Excel or Google Sheets) as an approach to MP5: use appropriate tools.

2 Active Note Taking and Explicit Instruction - 20 minutes

After the entry ticket, I review a Powerpoint presentation (Cinderella Slipper: Scatterplots, Residuals and Goodness of Fit) on fitting functions to data. I ask students to complete a number of Turn and Talks  (see strategy folder for more information) as part of this lesson. During this time students are also taking Two-Column Notes as a reference and resource.

This section continues to build off of the discussion from the entry ticket and focuses on having students think about the different types of functions we have learned about during the year and how to apply those functions to better explain and model data in contexts. I want students to be able to make connections from today's lessons to previous big ideas and concepts covered in other units.

3 Collaborative Work: Calculating and Plotting Residuals - 20 minutes

In this section we learn about what a residual is and go through an example from the GPA and Attendance data from the entry ticket as a class. Students also learn about how to plot and analyze residuals as a way to assess the fit of a particular function to data.  

I have students take the time to calculate all of the residuals for the entry ticket data for them to get a better understanding of how to calculate residuals but also think about what residuals are.

Please see the student worksheet (Classwork: Residuals Data Table) that teachers can print out and have students complete - this will save time so students do not have to copy the original data and can instead focus on the task at hand of analyzing and plotting residuals. 




Differentiating Instruction and Maintaining Access to Higher-Order Thinking Skills
Students with Disabilities

This video reflection is all about how I differentiate instruction to provide an opportunity to learn higher-level mathematical concepts to a wide range of students.


4 Collaborative Work: Creating a Line of Best Fit - 20 minutes

In this section I review different equations for determining a line of best fit for a data set. I pose the question, "Can you prove that the line of best fit provided is indeed THE line of best fit?" I judged this to be a manageable task for my students. During this section students work on the Collaborative Work: Creating a Line of Best Fit worksheet. 

I want students to compare linear function as a way of engaging in mathematical practices related to modeling bi-variate data with functions. I will actively engage with them as they complete this work. Without this support, some students may not be clear on what they are being asked to accomplished.  

*Note: I have found that many times I do not have time to complete this section in one 90 minute block. If class time runs out, this assignment could be used for homework, extra credit, and/or worked on the next day in class as an entry ticket.  

Adjusting Practice to Balance Support and Challenge
Adjustments to Practice

Reflecting on practice is a very important of teaching. In the middle of this particular lesson, I realized that I had placed too many expectations in terms of content for the class. I made the decision to not have students create a line of best fit for this lesson.

The reason for this decision is the lesson already involved a good deal of novel information and concepts for my students (residuals, calculating residuals, etc.) and I felt that many students were on the brink of becoming discouraged due to feeling overwhelmed with the math.

The common core asks teachers to provide opportunities for students to persevere in complex problems. I couldn't agree more with this goal. In the classroom, it can be a delicate balance between providing students with enough challenge (so they are engaged and motivated) with enough support (so that they do not stop working). I felt like making the decision to include the activity on creating a line of best fit for another day was a good example of striking this balance.

5 Recap and Homework: Our City Statistics Project - Creating Scatterplots - 20 minutes

With the remaining time in class, I ask students to begin work on their City Statistics Project. This is the culminating project/assessment for the unit, where students complete a cycle of inquiry into a research question related to their community. For more information on the project please see my Our City Statistics Lesson page. The Project: Our City Statistics Assignment Sheet for the project is also included in this section as a resource.


If the timing works, students will create scatter plots using the data from their projects. The reason for this is creating and interpreting scatter plots is highly connected to the day's objective. Alternatively, groups can utilize the remaining time in class to work on another section of the project, or even complete a section of the project for homework. 

There are a number of different ways technology can be incorporated into the lesson to support the scatterplot creations. Students can use a software like InspireData or Excel to easily create scatterplots. I have a class set of ipads and a Math Warehouse Scatterplot Maker website makes it easy for students to create and save scatterplots as a jpeg file. 

Technology as a Tool for Student Engagement
Student Ownership

In this reflection I discuss the evolution of the "Our City Statistics" Project and how technology has helped cultivate a class with increased student engagement and ownership.